Problem: A chemical is diluted out of a tank by pumping pure water into the tank and pumping the existing solution out of it, so the volume at any time $t$ is $20+2t$. The amount $z$ of chemical in the tank decreases at a rate proportional to $z$ and inversely proportional to the volume of solution in the tank. Which equation describes this relationship? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{dz}{dt}=-\dfrac{k(20+2t)}{z}$ (Choice B) B $\dfrac{dz}{dt}=k(20+2t)-\dfrac{1}{z}$ (Choice C) C $\dfrac{dz}{dt}=kz-\dfrac{1}{20+2t}$ (Choice D) D $\dfrac{dz}{dt}=-\dfrac{kz}{20+2t}$
Solution: The amount of chemical in the tank is denoted by $z$. The rate of change of the amount of the chemical is represented by $z'(t)$, or $\dfrac{dz}{dt}$. Saying that the rate of change is proportional to something means it's equal to some constant $k$ multiplied by that thing. That thing, in our case, is the current amount, $z$, of the chemical in the tank. We use a negative coefficient here to show that the change is removing the chemical, so we have $-kz$. Saying that the rate of change is inversely proportional to something means it's equal to some constant divided by that thing. That thing, in our case, is the volume, $20+2t$, of solution in the tank. In conclusion, the equation that describes this relationship is $\dfrac{dz}{dt}=-\dfrac{kz}{20+2t}$.